Oscillation-center theory for waves

Linear waves, both quantum and classical, can experience ponderomotive effects when propagating in modulated media. This phenomenon is analogous to the ponderomotive effect encountered by charged particles in high-frequency electromagnetic fields. Using the Weyl calculus and generalized Lie transformations for waves, we obtain a first-principle variational theory that describes the slowly-varying, oscillation-center dynamics of waves. In this approach, the characteristic wavelength of the modulations are allowed to be comparable to that corresponding to the wave. Quantum-like effects, such as the photon recoil effect, are retained. Examples and numerical results are presented. The theory is applied to several physical systems, such as: a Schr\"{o}dinger particle interacting with an oscillating electrostatic field, an electromagnetic (EM) wave propagating in a density-modulated plasma, and a Klein-Gordon particle interacting with arbitrary background and oscillatory EM fields. This work can serve as basis for future studies on the modulational instability.